Multidimensional quadratic BSDEs with separated generators

We consider multidimensional quadratic BSDEs with bounded and unbounded terminal conditions. We provide sufficient conditions which guarantee existence and uniqueness of solutions. In particular, these conditions are satisfied if the terminal condition or the dependence in the system are small enough.


Introduction
Backward stochastic differential equations (BSDEs) have been introduced by Bismut [1]. A BSDE is an equation of the form where W is a d-dimensional Brownian motion, ξ an n-dimensional random variable, called the terminal condition, and g : Ω × [0, T ] × R n × R n×d → R n is called the generator. The first existence and uniqueness result for BSDEs with an L 2 -terminal condition and a generator satisfying a Lipschitz growth condition is due to Pardoux and Peng [17]. In the following we restrict ourselves to the case where the generator satisfies some quadratic growth condition.
The first result on quadratic BSDEs is due to Kobylansky [15] who obtained solvability and a comparison theorem for the 1-dimensional (n = 1) equation with a bounded terminal condition and a generator g(s, y, z) satisfying a quadratic growth condition in z. Briand and Hu proved solvability of the 1-dimensional BSDE with unbounded terminal conditions in [5] and [6]. In [4] Briand and Elie introduced a method for 1-dimensional quadratic BSDEs with and without delay. Bahlali et al. [3] obtained solutions to the quadratic 1-dimensional case for generators which does not satisfy continuity assumptions and the terminal value does not have exponential moments. Several papers concern BSDEs with superquadratic BSDEs. Here, the first result is due to Delbaen et al. [9] who consider a generator which is convex in z and bounded terminal conditions. They showed that there exists a bounded terminal condition such that the associated BSDE does not admit any bounded solution and if the BSDE has a bounded solution, there exist infinitely many bounded solutions. Markovian superquadratic BSDEs with undbounded terminal conditions were studied in Masiero and Richou [16] and Richou [18]. In [7] Cheridito and Nam studied the case where the generator can grow arbitrarily fast in z and the terminal condition has bounded Malliavin derivative.
For multidimensional quadratic BSDEs there does not exist general existence results. Frei and dos Reis [11] and Frei [10] constructed counterexamples which show that multidimensional quadratic BSDEs may fail to have a global solution. A first solvability result for systems of quadratic BSDEs was obtained by Tevzadze [19] based on Banach's fixed point theorem. His result provides existence and uniqueness whenever the terminal condition is small. Cheridito and Nam [8] and Hu and Tang [12] obtained local solvability on [T − ε, T ] for some ε > 0 of systems of BSDEs with subquadratic generators and diagonally quadratic generators respectively, and both articles show that under an additional assumption on the generator the local solution can be extended to a global one. Cheridito and Nam [8] proved solvability for Markovian quadratic BSDEs via a linear growth FBSDE method and projectable quadratic BSDEs by using the 1-dimensional result of Kobylansky [15]. Frei [10] introduced the notion of split solution and studied the existence of multidimensional quadratic BSDEs by considering a special kind of terminal condition. In Bahlali et al. [2] existence is shown when the generator g(s, y, z) is strictly subquadratic in z and satisfies some monotonicity condition. Systems of quadratic BSDEs naturally arise in equilibrium pricing models in financial mathematics. Recently, Kramkov and Pulido [13] obtained a solvability result of systems of quadratic BSDE in a price impact model.
In this work we study the relationship between the size of the terminal condition and the degree of coupling of systems of quadratic BSDEs. We assume that the terminal condition is bounded and the generator depends on the value as well as on the control processes. We consider generators which can be separated into two parts: the coupled and the uncoupled part. We use the growth coefficients of the coupled part to characterize the degree of coupling. In the first step of the construction of the solution we view the uncoupled part as a parameter and solve a 1-dimensional quadratic BSDE (Lemma 3.1) the solvability result of which is based on Theorem 2 in [5], and provides an extension of Lemma 2.5 in [12]. Thus we can find a bounded set of candidate solutions where the value process is uniformly bounded and the control process is bounded in BMO. If the generator is independent of the value process, we can consider unbounded terminal conditions. In the second step we give two set of assumptions each of which make Banach's fixed point theorem applicable. For instance, Theorem 4.2 states that if the degree of coupling is small, the system is solvable for large terminal conditions. We show that our assumptions include examples of multidimensional quadratic BSDEs with bounded and unbounded terminal conditions which cannot be solved by the existing literature. For bounded terminal conditions, we exemplify this for generators which are the sum of squares. By Young's inequality we obtain similar existence results as in [12]. Moreover, our method extends the local solvability results of [8] and [12]. The additional assumption on generator needed in [8] in order to go from local to global works for our method as well.
The paper is organized as follows. In Section 2, we introduce some notations and collect some results from the literature. In Section 3, we state a result for one-dimensional BSDEs with parametrized generator. In Section 4, we prove our main existence results on multidimensional quadratic BSDEs. Finally, in Section 5 we discuss the connection to the existing literature and give examples.

Preliminaries
Fix a nonnegative real number T > 0, and let (W t ) t≥0 be a d-dimensional Brownian motion on a complete probability space (Ω, F , P ). Let (F t ) t≥0 be the augmented filtration generated by the Brownian motion W . We assume that F T = F and denote by P the predictable σ-algebra on Ω × [0, T ]. Inequalities and equalities between random variables and processes are understood in the P -almost sure and P ⊗ dt-almost sure sense, respectively. For two real numbers a, b ≥ 0, the minimum and maximum of a and b are denoted by a ∧ b and a ∨ b, respectively. The Euclidean norm is denoted by | · | and we denote by · ∞ the L ∞ -norm. The set of stopping times with values in [0, T ] is T .
For the sake of simplicity, we consider only the 2-dimensional quadratic BSDE. An extension to n-dimensions is straightforward.
where ξ i is an F T -measurable random variable and g i : and B(R 2×d ) denote the Borel sigma-algebra on R 2 and R 2×d , respectively. By a solution we mean a pair of predictable processes (Y, Z) such that (1) holds and t → Y t is continuous, t → Z t belongs to L 2 (0, T ) and t → g(t, Y t , Z t ) belongs to L 1 (0, T ) P -a.s.. Let S ∞ (R n ) denote the space of all n-dimensional continuous adapted processes such that For any uniformly integrable martingale M with M 0 = 0 and p ∈ [1, ∞), define We will use BM O p (P ) when it is necessary to indicate the underlying probability measure, and just write BM O when p = 2. For some adapted process b, satisfying the necessary integrability conditions, (b · W ) t∈[0,T ] denotes the stochastic integral process with respect to the Brownian motion W .
is a strictly positive martingale, and one obtains from Girsanov's theorem that dP dP = E T (b · W ) defines an equivalent probability measure under whichW t = W t − t 0 b s ds is a Brownian motion (Kazamaki [14]). For a 2d-dimensional process Z = (Z 1 , Z 2 ), we denote by Z · W := (Z 1 · W, Z 2 · W ) and the BMO-norm of Z ·W is given by We recall the following results from the literature.
(ii) If M BMO < 1, then for any stopping time τ ∈ T Lemma 2.2 For any p ∈ [1, ∞), there exists a constant L p > 0 such that for any uniformly integrable martingale M , it holds that The following lemma is based on Theorem 3.3 in [14]. We also refer the reader to Lemma 2.4 in [12] for a complete proof.

Lemma 2.3
For K > 0, there are constants c 1 > 0 and c 2 > 0 such that for any BMO martingale M , we have for any BMO martingale N such that N BMO(P ) ≤ K, whereM := M − M, N and dP dP : By Lemma 2.4 in [12], the constants in the previous lemma are given by where 1 p + 1 q = 1 and 1 p + 1 q = 1, C p and Cp are given by the reverse Hölder inequality, L 2q and L 2q are given by Lemma 2.2, and p,p are constants such that Φ(p) > K and Φ(p) >K, wherē K = 2(q − 1) log(C p + 1).
The reverse Hölder inequality is given in the following.
for any stopping time τ ∈ T with a constant C p depending only on p.

Auxiliary result for the one-dimensional BSDE
In order to state our main results, we first need the subsequent lemma on 1-dimensional BSDEs which are given by where f : We assume that the function f (ω, t, z) is continuous in z for P ⊗ dt-almost all (ω, t) ∈ Ω × [0, T ] and that there exist constants C > 0 and γ > 0 such that Moreover, we consider the following conditions: (A1) There exists a constant θ > 0 such that (ii) Assume that (A3) holds, then the BSDE (2) has a solution (Y, Z) such that Y is bounded and Z · W is a BMO martingale.
(iii) Assume that (A4) holds, then the BSDE (2) has a solution (Y, Z) such that Z · W is a BMO martingale and Moreover, suppose that (A1) holds, let (Ỹ ,Z) solve the BSDE (2) whereξ andf satisfy the set of conditions as in (ii) or (iii), andξ ≥ ξ andf ≥ f . Then it holds thatỸ t ≥ Y t . In particular, under (ii) and (iii) the solution is unique. Proof.
For n ≥ 1 let τ n be the stopping time Let u : R + → R + be given by Then u(| · |) is a C 2 -function on R, and by Itô's formula for all x ≥ 0 and by the growth condition on f and (A3) it holds that By taking conditional expectation with respect to F t on both sides of the previous inequality the last term vanishes whenever τ n ≤ t. On the complement t < τ n it is a martingale since it is a local martingale the quadratic variation process of which is bounded by definition. Hence we obtain a uniform norm of the left hand term for each n by (5) and (A3) and thus the dominated convergence theorem yields By taking the essential supremum over all stopping times in T to the left hand side of the previous inequality we get the the following BMO-bound on Z · W : (iii) By the martingale representation theorem, Combining the previous estimate with (3) we conclude thatŶ t : Applying Itô's formula to ϕ(|x|) = 1 (4γ) 2 (e 4γ|x| − 1 − 4γ|x|) and arguing as in (ii) (by using By a similar argument as in (ii) it holds that Z · W is a BMO martingale.
(iv) Let ∆Y :=Ỹ − Y and ∆Z :=Z − Z. Then We can find a predictable process b such that |b s | ≤ θ(1 + |Z s | + |Z s |) which already implies that b · W is a BMO martingale satisfying the following equation By taking conditional expectation with respect toP and F t on both sides of the previous equation it follows that ∆Y ≥ 0.
Corollary 3.2 If we replace in Lemma 3.1 in (ii) and (iii) the growth conditions on g by |g| ≤ |z| 2−κ where z is a BMO martingale and κ ∈ [0, 1), then the assertions remain valid.
Proof. By Young's inequality If z ·W BMO < B, by choosing λ =

Main existence results
The two dimensional system of BSDEs is given as follows: where the generator is of the form We consider the following conditions. If there is no risk of confusion we write y and z for the vectors (y 1 , y 2 ) and (z 1 , z 2 ), respectively. For each i = 1, 2, there are constants C, Theorem 4.1 Assume (B1)-(B5) and that (i) (β 1 + β 2 )T < 1, (ii) there exists δ ∈ (0, 1) such that 2 When the growth of generator is purely quadratic, C is allowed to be 0. where Then the system of BSDEs (6) has a unique solution (Y, Z) such that Proof. Fix y ∈ S ∞ (R 2 ) and let z · W is a BMO martingale with Define the function I mapping (y, z) to (Y, Z) where for each By Lemma 3.1 the 1-dimensional equation (6) admits a unique solution (Y i , Z i ) such that By Itô's formula, By (B2) and (B4) and since y and ξ i are bounded, By taking conditional expectation with respect to F t on both sides of the previous inequality and using the BMO-norm of z · W ,

Thus the set of candidate solutions is given by
Next we show that I : M → M mapping (y, z) → (Y, Z) is a contraction. Let (Y, Z) = I(y, z) and (Ȳ ,Z) = I(ȳ,z). Then by Itô's formula, If we take conditional expectation with respect to F t on both sides of the previous equation, the last term of the right hand side vanishes due to the martingale property. If in the first term on the right hand side we extract the uniform norm of (Y −Ȳ ) to the outside of the expectation and then by Young's inequality 2ab ≤ 1 4 a 2 + 4b 2 , we obtain it follows from Hölder's inequality and (a + b + c) 2 ≤ 3(a 2 + b 2 + c 2 )

BMO
By the Assumption (ii) the function I is a contraction. In the following theorem we obtain solvability of the system (6) under a slightly different set of assumptions.
Then the system of BSDEs (6) has a unique solution (Y, Z) such that Proof. In order to obtain the set of candidate solutions For the contraction argument we need to proceed differently. Let (Y, Z) = I(y, z) and (Ȳ ,Z) = I(ȳ,z). First taking square, second conditional expectation with respect to F t andP on both sides of the previous equality, and third by Hölder's inequality and 2ab ≤ a 2 + b 2 , By Lemma 2.3 there are two constants c 1 > 0 and c 2 > 0 which only depend on b · W BMO such that By a similar argument, we obtain By the Assumption (ii) the function I is a contraction.

Remark 4.3 By Young's inequality,
Thus whenever the coupled part includes subquadratic growth or has only subquadratic growth we have similar global solvability results by our method.
In the following two corollaries, the system of BSDEs is given by where ξ i and g i satisfy (B1)-(B5) while α 1 , α 2 , β 1 , β 2 = 0, that is the generator is independent of the value process while the system is fully coupled.

Corollary 4.5 Assume that
where 1 + e 4γ1CT (8γ 1 CT + 1) , and c 1 and c 2 only depend on D 1 and θ 1 ,c 1 andc 2 only on D 2 and θ 2 , and L is a fixed constant. Then the system of BSDEs (7) admits a unique solution (Y, Z) such that Y ∈ S ∞ (R 2 ) and Z · W BMO ≤ √ D 1 + D 2 .
In the next two corollaries the generators are independent of the value process but the system is only partially 3 coupled which has the consequence that the sets of candidates solutions for the control process can be separated. The system is given by where ξ i ∈ L ∞ (F T ), and for C, γ i ,γ i , θ i ,θ i > 0 the generator g i satisfies the following growth conditions Corollary 4.6 Assume additionally to the aforementioned conditions that there exists δ ∈ (0, 1) such that where Then the system of BSDEs (7) admits a unique solution (Y, Z) such that Y ∈ S ∞ (R 2 ) and Z i · W BMO ≤ √ D i for i = 1, 2.

Corollary 4.7 Assume that
where 1 + e 4γ1CT (8γ 1 CT + 1) , and c 1 and c 2 only depend on D 1 and θ 1 ,c 1 andc 2 only depend on D 2 and θ 2 , and L is a fixed constant. Then the system of BSDEs (7) admits a unique solution (Y, Z) such that Y ∈ S ∞ (R 2 ) and Proof. The growth conditions of the generators imply already (B2)-(B5), set α 1 = α 2 = β 1 = β 2 = 0 and replace z by z 1 and z 2 , respectively, in the proof of Theorem 4.2.

Discussion
Our results yield existence of some multidimensional quadratic BSDEs which cannot be solved by the existing literature. For the sake of illustration we consider the following system In this case the two main Theorems 4.1 and 4.2 can be summarized as follows, respectively.
(I) Given any θ 1 , ϑ 2 and any ξ 2 ∈ L ∞ , let ξ 1 and ϑ 1 , θ 2 be sufficiently small, then the BSDE (13) admits a unique solution (interchanging 1 and 2 in the previous statement maintains its truth). Indeed, we need to choose the growth coefficients and the terminal condition according to the following scheme.
(Ia) Let ξ 1 = 0. Choose δ > 0 and θ 2 > 0 to be sufficiently small such that (Ib) Next choose ϑ 1 to be small enough such that For instance, the following choice satisfies the previous conditions: (II) Given any θ 1 , θ 2 and any ξ 1 , ξ 2 ∈ L ∞ (F T ), let ϑ 1 and ϑ 2 be sufficiently small, then the BSDE (13) admits a unique solution. Indeed, choose the growth conditions and the terminal conditions according to the following scheme.
(IIa) Choose ϑ 1 > 0 and ϑ 2 > 0 to be small enough such that (IIb) Since θ 1 and θ 2 and ξ 1 and ξ 2 are given, c 1 , c 2 ,c 1 ,c 2 are fixed constants, it remains to choose ϑ 2 and ϑ 1 to be sufficiently small such that In (I) and (II) we are allowed to have large coefficients and large terminal conditions which cannot be solved by Tevzadze's result [19].

Local solutions
By restricting the time interval to [T − ε, T ] for some ε > 0 in Theorem 4.1 and 4.2, respectively, we obtain the following local solvability results related to the recent works of Cheridito and Nam [8] and Hu and Tang [12].
Then the system of BSDEs (6) admits a unique solution (Y, Z) on [T − ε, T ] such that Moreover, whenever the coupled part only has subquadratic growth condition, by Young's inequality If z · W BMO ≤ B, by choosing λ = whereby c 1 and c 2 only depend on D 1 and θ 1 ,c 1 andc 2 only on D 2 and θ 2 , and L is a fixed constant.
We can always find a pair of (ε, B) if we let ε be sufficiently small guaranteeing the local solvability of multidimensional BSDEs with diagonally quadratic generator as in Hu and Tang [12]. However, our bounds A and √ D 1 + D 2 are different. If in addition, for i = 1, 2, the generator g i satisfies g i (t, y, z) = F i (t, z) + H i (t, y, z) where |F i (t, z)| ≤ C i |z i ||z| and |H i (t, y, z)| ≤ C i (1 + |y| + |z|). Then by Lemma 4.3 in Cheridito and Nam [8], there exists a constant K such that |Y t | ≤ K, and replacing ξ 1 ∞ + ξ 2 ∞ by 2K in the definition of A, we can extend the above local result to the global one recursively.

Unbounded terminal conditions
When the generator is independent of the value processes, we can consider unbounded terminal condition as in Lemma 3.1. By the martingale representation theorem it holds that Applying Itô's formula to ϕ(|x|) = |x| 2 and ϕ(|x|) = 1 (4γ) 2 (e 4γ|x| − 1 − 4γ|x|) and by arguing similarly as in the Theorems 4.1 and 4.2, respectively. 4 We obtain similar results as the ones we obtained for the bounded case, both for global and local solutions.